036i - Creating Expressions from Context

Lesson 36 — Problem Solving: Creating and Simplifying Expressions from Contexts

Strand: Algebra | Descriptor: AC9M8A01 | Duration: 45 minutes

Learning Intentions

• To translate worded and real-world contexts into linear algebraic expressions.
• To expand, simplify or factorise a created expression as needed to answer a question.

Success Criteria

I can:

1. Define a variable clearly before writing an expression.
2. Translate a worded description into an algebraic expression.
3. Simplify or expand a created expression to make it more useful.
4. Use a created expression to answer a question by substitution.
5. Justify each stage of my reasoning using Polya’s problem-solving cycle.

Warmup

(5 minutes — quick translation drill, mini whiteboards)

Write an expression for each, using for the unknown number.

1. Five more than a number.
2. Triple a number, then subtract four.
3. A number decreased by seven, then doubled.
4. The sum of a number and its double.

Answers: ; ; ; .

Discussion point: Q3 and Q4 both need care with order — “decreased by seven, then doubled” is not the same as “doubled, then decreased by seven.” Compare with using : versus .

Activities

Activity 1 — Explicit Instruction: the Translation Protocol (12 min)

The four-step protocol for context problems:

1. Define the variable in words, e.g. “let = the number of…”
2. Translate each piece of the context into algebra.
3. Combine/simplify — expand brackets and collect like terms if needed.
4. Use the expression — substitute values, or set it up ready for further work.

I do — a hire cost problem. A kayak hire company charges a 15$6$ per hour.

Find the cost for hours: , so 39$.

I do — a perimeter-and-cost problem requiring simplification. A rectangular garden bed has width metres and length metres. Fencing costs 12$ per metre.

Check at : m; cost ; via the expanded formula: ✓

We do: A taxi charges a 4$2.50k12$ km.

Activity 2 — Independent Practice: Building and Simplifying (10 min)

You do: For each context, define your variable, write an expression, simplify if needed, then answer the question.

1. A pizza shop charges 8$3p5$ pizzas.
2. A rectangular pool has width and length . Write a simplified expression for its perimeter.
3. Three consecutive whole numbers start at . Write a simplified expression for their sum.
4. A phone plan costs 25$0.40m30$ extra minutes.

(Answers: 1. ; at : 43P=2x+2(2x-1)=6x-2n+(n+1)+(n+2)=3n+3C=25+0.4mm=30$37$.)

Activity 3 — Inquiry: Designing a Fair Pricing Plan (18 min)

Pairs, then whole-class share. This is the lesson’s main applied task.

A school fete runs a “mystery box” stall. Two pricing plans are proposed:

Plan A: a 5$2$3$ per box opened, with the first box free.

1. Define a variable and write a simplified expression for the total cost under each plan.
2. For how many boxes are the two plans equal in cost?
3. A customer plans to open boxes. Which plan is cheaper, and by how much?
4. The stall organiser wants to redesign Plan B so it is never more expensive than Plan A, no matter how many boxes are opened. Investigate: is this possible by changing only the price per box?

Socratic scaffolding for Q4 (the harder question):

Prompt	Purpose
Understand the problem.	You need Plan B’s total cost to be Plan A’s cost for every whole-number number of boxes, not just some.
What are the two expressions?	Plan A: . Plan B (with first box free): for .
Devise a plan.	Expand and compare the two expressions directly, focusing on the coefficient of (the “per-box” rate) and the constant (the “starting” cost).
Carry out the plan.	. Compare with : the difference is .
Interpret.	For , (Plan B cheaper); for , (Plan A cheaper); equal exactly at .
Looking back — so is “never more expensive” achievable by changing the price per box alone?	Only if the new per-box rate for Plan B is no greater than Plan A’s rate of 2n$2C_B=2(n-1)=2n-2$7C_A=2n+5n$.

Answers: 1. ; ; 2. equal when boxes; 3. at : , — Plan B is 2$2n$2C_B=2n-2$) makes it never more expensive.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Write an expression for “four less than triple a number .”
2. A car wash costs 6$4s3$ extra services.
3. A rectangle has width and length . Write a simplified expression for its perimeter.
4. Reasoning. Two gym plans cost and , where is visits. For how many visits are the plans equal? Which plan is cheaper for visits?
5. Explain, in one sentence, why defining your variable clearly is an essential first step in these problems.

Answers: 1. ; 2. ; at : 18P=4x+1020+10v=15v \Rightarrow v=4v=10C_1=120C_2=150$ — Plan 1 cheaper; 5. without a clear definition, the meaning of the expression (and any answer derived from it) is ambiguous and cannot be checked or communicated.

Common Misconceptions

Misconception	How to pre-empt it
Skipping the “define the variable” step, leading to expressions that don’t match the context.	Insist on a written “let = …” line before any algebra, every time.
Translating “decreased by, then doubled” the wrong way round, e.g. instead of .	Read the phrase slowly, translating one operation at a time in the order given.
Forgetting to simplify a created expression before using it, leading to error-prone repeated substitution.	Model why a simplified expression () is faster and safer to substitute into than the unsimplified original.
Assuming the plan with the lower starting fee is always cheaper.	Activity 3 confronts this directly — the rate per unit matters more for large quantities than the starting fee.
Believing a “fair” comparison only needs one test value.	Reinforce from Lesson 32: agreement at one value doesn’t mean agreement everywhere — compare simplified expressions or test multiple values.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A café charges 4$2.50cb35$ biscuits.

Answer

24.50$.

E2 (AMC Junior style). Two water tanks are being filled. Tank A starts with L and fills at L/min. Tank B starts empty and fills at L/min. Write expressions for both, then find when they contain equal amounts of water.

Answer

After minutes.

E3 (Challenge). A rectangle’s perimeter is described by two different builders: Builder 1 says ; Builder 2 says . Are they describing the same rectangle? Justify with full working.

Answer

Yes — Builder 1’s expression simplifies to exactly Builder 2’s, so they describe the same rectangle for every .

E4 (Investigation). A ride-share app charges (km ), while a taxi charges . Find the break-even distance, and explain, using the coefficients, which is cheaper for very long trips.

Answer

Break-even at km. For long trips, the taxi’s lower per-km rate ( vs ) means it becomes cheaper beyond km.

Homework

1. Write a simplified expression for each: (a) six more than double a number (b) a number decreased by three, then tripled (c) the sum of two consecutive numbers starting at .
2. A plumber charges a 50$70h3.5$-hour job.
3. A rectangular pool has width and length . Write a simplified expression for (a) its perimeter (b) its area.
4. Two market stalls charge as follows: Stall A: 2$1.50$2$ per item. Write expressions for both, and find the number of items at which the costs are equal.
5. Reasoning. Explain, using Activity 3’s plans, why a lower “starting fee” doesn’t guarantee a plan is cheaper overall.
6. Reasoning. A student writes ” less than double ” as . Explain the error and give the correct expression.
7. Challenge. A school printing budget allows 40812p$, and find the number of pages at which the costs are equal.
8. Challenge. Show that the two perimeter expressions and describe the same rectangle, with full working.

Answers: Q1 — (a) (b) (c) . Q2 — ; at : 295P=4x+8A=x^2+4xC_A=2+1.5nC_B=2n2+1.5n=2n \Rightarrow n=4n=8n52n-5C_1=40+0.08pC_2=0.12p40+0.08p=0.12p \Rightarrow 0.04p=40 \Rightarrow p=10002(3x+2)+2(x+1) = 6x+4+2x+2 = 8x+6$, matching exactly.